Why Reprice a Portfolio Under Yield-Curve Scenarios?
A yield curve can move in several ways. A parallel shift changes rates at all maturities by the same amount; a steepener changes longer-maturity rates more than shorter-maturity rates; and a flattener narrows the difference between short- and long-maturity rates. These scenarios can affect a portfolio differently even when their average rate changes look similar.
Repricing makes the impact concrete: value the same fixed-income positions under a baseline curve and a specified scenario curve, then compare the results. For fixed cash flows, direct cash-flow repricing gives the value under the stated curves. Key rate sensitivities provide a faster, first-order estimate when full repricing is unavailable or when the scenario is summarized by curve-node changes.
Cash-Flow Repricing
For a fixed-cash-flow instrument, each payment is discounted using the scenario spot rate for its payment date. Assume annual effective spot rates \(s_t\), annual compounding, and known cash flows \(C_t\) payable at the end of year \(t\). Its value is
A scenario is applied by replacing each baseline spot rate with its scenario value, then recalculating the present value. This method assumes the cash flows do not change with rates. If payments depend on an embedded option or other rate-sensitive behavior, that behavior must also be modeled; simply discounting unchanged payments is not a complete repricing.
Worked Example: Exact Repricing Under a Parallel Shift
A hypothetical portfolio holds two zero-coupon bonds. One pays $100 at the end of year 1, and the other pays $200 at the end of year 2. The annual effective spot curve is flat at 5.00%. Reprice the portfolio after a parallel increase of 100 basis points. Report dollar values to the nearest cent.
Method. Discount each payment at its maturity-matched spot rate, first under the baseline curve and then under the scenario curve. The scenario curve is flat at 6.00%.
Baseline value. The year-1 discount factor is \(1/1.05=0.952381\), so that payment is worth \(100/1.05=\$95.24\). The year-2 discount factor is \(1/1.05^2=1/1.1025=0.907029\), so that payment is worth \(200/1.1025=\$181.41\). Thus \(V_0=\$95.24+\$181.41=\$276.64\), using unrounded values in the sum.
Scenario value. At 6.00%, the year-1 discount factor is \(1/1.06=0.943396\), giving \(100/1.06=\$94.34\). The year-2 discount factor is \(1/1.06^2=1/1.1236=0.889996\), giving \(200/1.1236=\$178.00\). Thus \(V_s=\$272.34\).
Change and check. Using unrounded values, \(\Delta V=V_s-V_0=\$272.3389-\$276.6440=-\$4.3051\), or a loss of \(\$4.31\). The sign is consistent with fixed positive cash flows losing value when their discount rates rise.
The common mistake is to apply the 100-basis-point shift as 100, or even as 0.10, in the rate formula. One hundred basis points is 1.00 percentage point, so the decimal rate change is \(0.01\).
Key Rate Sensitivities for Nonparallel Scenarios
A portfolio's key rate duration measures its relative value sensitivity to a change at a particular maturity point on the curve, with the curve's other movements specified by the chosen bump convention. Write \(KRD_j\) for the key rate duration at node \(j\), and \(\Delta y_j\) for the scenario change at that node, expressed as a decimal. For sufficiently small changes, the first-order estimate is
For example, a rate increase of 25 basis points is \(\Delta y=0.0025\), not \(25\) or \(0.25\). A positive key rate duration combined with a rate increase contributes a negative estimated value change. A rate decrease contributes a positive one.
The approximation follows from a first-order change in value. If \(V\) depends on node yields \(y_1,\ldots,y_m\), its local change is approximately \(\Delta V\approx\sum_j(\partial V/\partial y_j)\Delta y_j\). Key rate duration is defined by \(KRD_j=-(1/V_0)(\partial V/\partial y_j)\), using the applicable node-bump convention. Substituting that definition gives the displayed estimate. The approximation omits higher-order effects, so it is not identical to full repricing, particularly for large shifts or material curvature.
Worked Example: Comparing a Steepener and a Flattener
A hypothetical $10,000,000 portfolio has key rate durations of 1.2 at the 2-year node, 4.0 at the 5-year node, and 2.5 at the 10-year node. Estimate its value change under two scenarios: (a) a steepener with node changes of +25, +75, and +125 basis points; and (b) a flattener with changes of +75, +25, and −25 basis points. Use the first-order key rate estimate.
Method. Convert each node change to decimal form, multiply by the corresponding key rate duration, sum the contributions, and multiply by negative portfolio value.
(a) Steepener. The changes are \(0.0025\), \(0.0075\), and \(0.0125\). The weighted sum is \(1.2(0.0025)+4.0(0.0075)+2.5(0.0125)=0.003+0.030+0.03125=0.06425\). Therefore, \(\Delta V\approx-\$10{,}000{,}000(0.06425)=-\$642{,}500\). The estimated scenario value is \(\$10{,}000{,}000-\$642{,}500=\$9{,}357{,}500\).
(b) Flattener. The changes are \(0.0075\), \(0.0025\), and \(-0.0025\). The weighted sum is \(1.2(0.0075)+4.0(0.0025)+2.5(-0.0025)=0.009+0.010-0.00625=0.01275\). Thus \(\Delta V\approx-\$10{,}000{,}000(0.01275)=-\$127{,}500\), and the estimated scenario value is \(\$9{,}872{,}500\).
Comparison and check. Both scenarios produce estimated losses for this portfolio, but the steepener's loss is larger by \(\$642{,}500-\$127{,}500=\$515{,}000\). The negative long-node change in the flattener offsets part of the loss from rising short- and intermediate-node rates. The common mistake is to treat every rate move as the same parallel shift, discarding the node-specific exposure that distinguishes the scenarios.
Exact Repricing of a Nonparallel Scenario
When scenario spot rates are available and cash flows are fixed, direct repricing avoids relying on a linear sensitivity approximation. The following example applies a steepener directly to a small cash-flow portfolio.
Worked Example: Direct Repricing Under a Steepener
A hypothetical portfolio pays $100,000 at the end of year 1 and $100,000 at the end of year 2. The baseline annual effective spot curve is flat at 2.00%. Under a steepening scenario, the 1-year spot rate rises to 3.00% and the 2-year spot rate rises to 4.00%. Calculate the exact value change, with dollar amounts rounded to the nearest cent.
Method. Discount each payment at its own maturity spot rate under both curves. The scenario changes are +100 basis points at 1 year and +200 basis points at 2 years.
Baseline value. The 1-year payment has present value \(100{,}000/1.02=\$98{,}039.22\). For year 2, \(1.02^2=1.0404\), so the payment has present value \(100{,}000/1.0404=\$96{,}116.88\). Using unrounded amounts, \(V_0=\$194{,}156.09\).
Scenario value. The 1-year payment is worth \(100{,}000/1.03=\$97{,}087.38\). The year-2 discount denominator is \(1.04^2=1.0816\), so that payment is worth \(100{,}000/1.0816=\$92{,}455.62\). Using unrounded amounts, \(V_s=\$189{,}543.00\).
Change and check. The exact change is \(\Delta V=\$189{,}543.00-\$194{,}156.09=-\$4{,}613.09\), using unrounded values before rounding. Both rates rise, so the value of these fixed positive payments falls. This check would fail if the scenario value were reported above the baseline value.
A common mistake is to discount both payments at the 1-year rate, or to apply the 2-year rate only once to the year-2 payment. Each payment uses its maturity-matched spot rate, raised to the number of years to payment under annual compounding.
Choosing and Interpreting a Repricing Method
Use a method that matches the information available and the question being asked. If scenario spot curves and fixed cash flows are known, cash-flow repricing gives the value under those specified curves. If the scenario is given as changes at selected curve nodes and the relevant key rate durations are known, the first-order estimate gives a quick, transparent comparison. These two methods serve related but distinct purposes: one calculates scenario values from cash flows, while the other approximates a value change from local sensitivities.
| Scenario or method | What changes | Practical implication |
|---|---|---|
| Parallel shift | Every curve node moves by the same amount | Aggregated rate sensitivity can summarize the first-order effect |
| Steepener | Longer rates rise more than shorter rates, or longer rates fall less | Long-maturity exposures may dominate the effect |
| Flattener | The gap between long and short rates narrows | Short- and long-node contributions can offset |
| Cash-flow repricing | Scenario spot rates are applied to each fixed payment date | Calculates scenario value under the stated curve assumptions |
| Key rate estimate | Node changes are combined with key rate durations | Provides a local, first-order estimate rather than exact repricing |
A useful control is to state the scenario changes in basis points and decimal form before calculating. Another is to check the direction of the result against the cash flows and rate changes. For fixed, positive payments, an increase in all relevant spot rates lowers present value; a decrease raises it. With mixed rate movements, the net result depends on the portfolio's exposure at each affected maturity, so the sign should be checked against the weighted contributions rather than guessed from the scenario label.
Key rate estimates are most informative when the shocks are modest and the portfolio response is locally close to linear. For larger shocks, substantial curvature, changing cash flows, or option-embedded positions, use a suitable full valuation approach rather than treating a first-order estimate as an exact scenario value. A scenario gain or loss is also not, by itself, a probability statement: it describes the effect of the specified curve movement, not how likely that movement is.
Check Your Understanding
For each problem, identify the repricing method and keep the yield-change units consistent. Dollar answers are rounded to the nearest cent where needed.
- A hypothetical bond pays $20,000 in one year. Its baseline annual effective spot rate is 4.00%; the scenario rate is 5.00%. What is the exact change in value? (Answer: −$183.15)
- A $5,000,000 portfolio has key rate durations 1.5 at 2 years and 6.0 at 10 years. The corresponding rates rise 10 basis points and fall 20 basis points. Estimate the value change. (Answer: +$52,500)
- A portfolio worth $8,000,000 has total parallel-shift duration of 5.0. Estimate its value change under a parallel decrease of 40 basis points. (Answer: +$160,000)
- A $4,000,000 portfolio has key rate durations 0.8 at a short node and 3.0 at a long node. The short rate rises 50 basis points while the long rate falls 20 basis points. Estimate the value change. (Answer: +$8,000)
- A fixed payment of $1,000 is due in two years. The annual effective spot rate is 3.00% under the baseline curve and 4.00% under the scenario curve. What is the exact loss from repricing? (Answer: $18.04)